Two regularization methods for a Cauchy problem for the Laplace equation. (English) Zbl 1132.35493

Summary: A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for \(y=0\), and boundary data are prescribed for \(x=0\) and \(x=\pi \). The solution is sought for \(0<y\leqslant 1\). We propose two different regularization methods for the ill-posed problem based on separation of variables. Both methods are applied to find regularized solutions which are stably convergent to the exact one with explicit error estimates.


35R25 Ill-posed problems for PDEs
35J25 Boundary value problems for second-order elliptic equations
35B35 Stability in context of PDEs
35A35 Theoretical approximation in context of PDEs
Full Text: DOI


[1] Beck, J. V.; Blackwell, B.; Clair, C. R., Inverse Heat Conduction. Ill-Posed Problems (1985), John Wiley & Sons: John Wiley & Sons New York · Zbl 0633.73120
[2] Cannon, J. R., Error estimates for some unstable continuation problems, J. Soc. Industr. Appl. Math., 12, 270-284 (1964) · Zbl 0134.08501
[3] Cannon, J. R.; Miller, K., Some problems in numerical analytic continuation, SIAM Numer. Anal., 2, 87-98 (1965) · Zbl 0214.14805
[4] Charton, M.; Reinhardt, H.-J., Approximation of Cauchy problems for elliptic equations using the method of lines, WSEAS Trans. Math., 4/2, 64-69 (2005) · Zbl 1205.65303
[5] Douglas, J., A numerical method for analytic continuation, (Boundary Value Problems in Difference Equation (1960), University of Wisconsin Press: University of Wisconsin Press Madison, WI), 179-189 · Zbl 0100.12405
[6] Eldén, L., Approximations for a Cauchy problem for the heat equation, Inverse Problems, 3, 263-273 (1987) · Zbl 0645.35094
[7] Eldén, L., Hyperbolic approximations for a Cauchy problem for the heat equation, Inverse Problems, 4, 59-70 (1988) · Zbl 0697.35060
[8] Eldén, L.; Berntsson, F.; Regińska, T., Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. Sci. Comput., 21, 6, 2187-2205 (2000) · Zbl 0959.65107
[9] Engl, H. W.; Hanke, M.; Neubauer, A., Regularization of Inverse Problems (1996), Kluwer Academic Publishers: Kluwer Academic Publishers Boston, MA · Zbl 0859.65054
[10] Falk, R. S., Approximation of inverse problems, (Colton, D.; Ewing, R.; Rundell, W., Inverse Problems in Partial Differential Equations (1990), SIAM: SIAM Philadelphia, PA), 7-16 · Zbl 0706.65124
[11] Falk, R. S.; Monk, P. B., Logarithmic convexity for discrete harmonic functions and the approximation of the Cauchy problem for Poisson’s equation, Math. Comp., 47, 135-149 (1986) · Zbl 0623.65095
[12] Hadamard, J., Lectures on the Cauchy Problem in Linear Differential Equations (1923), Yale University Press: Yale University Press New Haven, CT · JFM 49.0725.04
[13] Han, H., The finite element method in a family of improperly posed problems, Math. Comp., 38, 55-65 (1982) · Zbl 0476.65064
[14] Han, H.; Reinhardt, J.-J., Some stability estimates for Cauchy problems of elliptic equations, J. Inverse Ill-Posed Probl., 5, 437-454 (1997) · Zbl 0895.35105
[15] Hào, D. N., A mollification method for ill-posed problems, Numer. Math., 68, 469-506 (1994) · Zbl 0817.65041
[16] Hao, D. N.; Hien, P. M., Stability results for the Cauchy problem for the Laplace equation in a strip, Inverse Problems, 19, 833-844 (2003) · Zbl 1046.35010
[17] Hao, D. N.; Van, T. D.; Gorenflo, R., Towards the Cauchy problem for the Laplace equation, (Banach Center Publ., vol. 27 (1992)), 111-128 · Zbl 0820.35032
[18] Kabanikhin, S. I.; Karchevsky, A. K., Optimizational method for solving the Cauchy problem for an elliptic equation, J. Inverse Ill-Posed Probl., 3, 21-46 (1995) · Zbl 0833.65107
[19] Kirsch, A., An Introduction to the Mathematical Theory of Inverse Problems (1996), Springer-Verlag: Springer-Verlag Berlin · Zbl 0865.35004
[20] Kubo, M., \(L^2\)-conditional stability estimate for the Cauchy problem for the Laplace equation, J. Inverse Ill-Posed Probl., 2, 253-261 (1994) · Zbl 0817.35121
[21] Kubo, M.; Iso, Y.; Tanaka, O., Numerical analysis for the initial value problem for the Laplace equation, (Tanaka, M.; Du, Q.; Honma, T., Boundary Element Methods (1993), Elsevier: Elsevier Amsterdam), 337-344
[22] Qian, Z.; Fu, C.-L.; Xiong, X.-T., A modified method for a non-standard inverse heat conduction problem, Appl. Math. Comput., 180, 453-468 (2006) · Zbl 1105.65097
[23] Regińska, T.; Regiński, K., Approximate solution of a Cauchy problem for the Helmholtz equation, Inverse Problems, 22, 975-989 (2006) · Zbl 1099.35160
[24] Reinhardt, J.-J.; Han, H.; Hào, D. N., Stability and regularization of discrete approximation to the Cauchy problem for the Laplace’s equation, SIAM Numer. Anal., 36, 890-905 (1999) · Zbl 0928.35184
[25] Trong, D. D.; Long, N. T.; Dinh Alain, P. N., Nonhomogeneous heat equation: Identification and regularization for the inhomogeneous term, J. Math. Anal. Appl., 312, 93-104 (2005) · Zbl 1087.35095
[26] Trong, D. D.; Quana, P. H.; Dinh Alainb, P. N., Determination of a two-dimensional heat source: Uniqueness, regularization and error estimate, J. Comput. Appl. Math., 191, 50-67 (2006) · Zbl 1096.65097
[27] Weber, C. F., Analysis and solution of the ill-posed inverse conduction problem, Int. J. Heat Mass Transf., 24, 1783-1792 (1981) · Zbl 0468.76086
[28] Qian, Z.; Fu, C.-L., Regularization strategies for a two-dimensional inverse heat conduction problem, Inverse Problems, 23, 1053-1068 (2007) · Zbl 1118.35073
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.